The nth term of a sequence: finding the rule and any term

A sequence is a list of numbers that follows a rule, such as 5, 9, 13, 17. The nth term is a short rule that gives any number in the list without counting through all the others.

Using a rule

Suppose the rule is 4n + 1. The letter n stands for the position of the term: 1st, 2nd, 3rd and so on. Put the position in place of n.

• 1st term: 4 × 1 + 1 = 5
• 2nd term: 4 × 2 + 1 = 9
• 3rd term: 4 × 3 + 1 = 13
• 10th term: 4 × 10 + 1 = 41

Do the multiplication first, then the adding (the order of operations, see BODMAS).

Finding the rule

For a sequence that goes up by the same amount each time, follow three steps.

1. Find the common difference between neighbours. In 5, 9, 13, 17 it is 4.
2. The rule starts with that number times n: 4n.
3. Compare 4n with the real term. For n = 1, 4n gives 4, but the real first term is 5. You need +1. The rule is 4n + 1.

More worked examples

1. Multiples. 3, 6, 9, 12 goes up by 3 and 3 × 1 = 3 already matches the first term. The rule is 3n, so the 50th term is 150.

2. Rule with a subtraction. For 2, 7, 12, 17 the difference is 5. 5n gives 5 for n = 1, but the first term is 2, so subtract 3. The rule is 5n − 3.

3. A term from a rule. The nth term is 3n + 2. The 8th term is 3 × 8 + 2 = 26.

4. Is a number in the sequence? Is 100 in the sequence 4n + 1? Solve 4n + 1 = 100: 4n = 99, so n = 24.75. Positions are whole numbers, so 100 is not in the sequence.

5. A sequence that goes down. 20, 17, 14, 11 goes down by 3. The rule is 23 − 3n: for n = 1 it gives 20.

Mistakes to avoid

• Using the first term as the rule. The first term is 5, but the rule is 4n + 1. The difference tells you the number with n.
• Adding before multiplying. For 4n + 1 at n = 10, do 4 × 10 first (40), then add 1.
• Forgetting to check. Put n = 1 and n = 2 into your rule: both should match.

Practise it

• Nth term practice online with instant feedback and a mastery score.
• Nth term worksheets with an answer key.
• Number Sequences: a game about spotting patterns.

Common questions

What does n mean in the nth term?

It is the position in the sequence: n = 1 is the first term, n = 2 the second, and so on.

How do I find the 100th term?

Put n = 100 into the rule. For 4n + 1 that is 4 × 100 + 1 = 401.

What if the difference is not the same each time?

Then it is not a simple (linear) sequence. Squares (1, 4, 9, 16) follow n², and other patterns need other rules.

Which grade learns this?

Usually Grade 6 to Grade 8 (Class 6 to 8) as an introduction to algebra.

This guide is general information. For exams, follow your school, teacher and official exam board, and use your own judgement about what suits your child.